Identity and Search in Social Networks D.J.Watts, P.S. Dodds, M.E.J. Newman

1. Maryam Fazel-Zarandi Identity and SearchinSocial NetworksD.J.Watts, P.S. Dodds, M.E.J. Newman

2. Outlines • Introduction • The Hierarchical Model • Discussion

3. Introduction

4. Source Milgram’s Experiment • Short chains of acquaintances exist. • People are able to findthese chains using only local information.

5. Results in Literature • Connected random networks have short average path lengths: xij log(N) • N = population size, • xij = distance between nodes i and j.

6. Results in Literature • Kleinberg (2000) demonstrated that emergence of the second phenomenon requires special topological structure. • For each node i: • local edges d(i,j) ≤ p • long-range directed edges to q random nodes Pr(ij) ~ d(i,j)-a

7. Results in Literature • If networks have a certain fraction of hubs can also search well. • Basic idea: get to hubs first • Hubs in social networks are limited.

8. The Hierarchical Model

9. Hierarchical Model – Why? How? • Basic idea: impose some high-level structure, and fill in details at random. • Incorporate identity. • Need some measure of distance between individuals. • Some possible knowledge: • Target's identity, friends' identities, friends' popularity, where the message has been.

10. Hierarchical Network Construction • xij = the height of the lowest common ancestor level between i and j • z connections for each node with probability: p(x) = ce-αx Network constructed from template Hierarchical template for the network

11. Hierarchical Network Construction • Individuals hierarchically partition the social world in more than one way. • h = 1, …, H hierarchies • Identity vector • is position of node i in hierarchy h. • Social distance:

12. Directing Messages • At each step the holder i of the message passes it to one of its friends who is closest to the target t in terms of social distance. • Individuals know the identity vectors of: • themselves, • their friends, • the target.

13. Expected Number of Steps • What is the expected number of steps to forward a message from a random source to a random target? • Define q as probability of an arbitrary message chain reaching a target. • Searchable network: Any network for which q≥ r for a desired r.

14. Number of Steps - Results • If message chains fail at each node with probability p, require where L = length of message chain. • Approximation: L ln r / ln (1 - p) q = (1 - p)L ≥ r

15. Searchable Network Regions • In H-αspace • p = 0.25, r = 0.05 • b = 2 • g = 100, z = 99 • N=102400 • N=204800 • N=409600

16. Probability of Message Completion • α = 0 (squares) versus α= 2 (circles) • N = 102400 • q ≥ r q < r r = 0.05

17. Milgram's Data • N = 108 • b = 10 • g = 100 • z = 300 • Lmodel 6.7 • Ldata 6.5 • α = 1, H = 2

18. Discussion

19. Is this an acceptable model? • Simple greedy algorithm. • Represents properties present in real social networks: • Considers local clustering. • Reflects the notion of locality. • High-level structure + random links.

20. Can the Model be Extended? • Should we consider other parameters such as friend’s popularity information in addition to homophily? • Allow variation in node degrees? • Assume correlation between hierarchies? • Are all hierarchies equally important?

21. Applications • Can solutions to sociology problems inform other areas of research? • Suggested applications: • Construction of peer-to-peer networks. • Search in databases.

22. Thank You!Any Questions???